Fourier Series - Sawtooth Wave

Program 1

Sawtooth Wave

Python 3.11 Ready

import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
from scipy.integrate import quad

x_ = np.linspace(-20, 20, 10000)
T = 10  # Time period
n2 = 10  # Number of Harmonics
n1 = n2 + 1

# Define the sawtooth wave function
def f(x):
    return signal.sawtooth(2 * np.pi * (1/T) * x)

# Define a0
a0, _ = quad(lambda x: f(x), -T/2, T/2)
a0 = (2/T) * a0
print('a0', a0)

# Define an, bn
def integrand_an(x, n):
    return f(x) * np.cos(n * 2 * np.pi * x / T)

def integrand_bn(x, n):
    return f(x) * np.sin(n * 2 * np.pi * x / T)

anm = [(2/T) * quad(lambda x: integrand_an(x, i), -T/2, T/2)[0] for i in range(0, n1)]
bnm = [(2/T) * quad(lambda x: integrand_bn(x, i), -T/2, T/2)[0] for i in range(0, n1)]

print('an', anm)
print('bn', bnm)

# Fourier Series function
def fs(x):
    sum = a0 / 2
    for i in range(1, n1):
        sum += bnm[i] * np.sin(i * 2 * np.pi * x / T) + anm[i] * np.cos(i * 2 * np.pi * x / T)
    return sum

y = np.array([f(i) for i in x_])
fss = np.array([fs(i) for i in x_])

plt.plot(x_, y, color="blue", label="Signal")
plt.plot(x_, fss, color="red", label="Fourier series approximation")
plt.title("Fourier Series approximation number of harmonics: " + str(n2))
plt.legend()
plt.savefig('plot.png')
plt.show()

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

Fourier series approximation of a sawtooth wave signal

Program 2

Sawtooth Wave

Python 3.11 Ready

import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
from scipy.integrate import quad

x_ = np.linspace(-20, 20, 10000)
T = 10  # Time period
n2 = 10  # Number of Harmonics
n1 = n2 + 1

# Define the sawtooth wave function
def f(x):
     # Normalize to [0, T)
    x = (x % T) / T
    return 2 * (x - np.floor(x + 0.5))

# Define a0
a0, _ = quad(lambda x: f(x), -T/2, T/2)
a0 = (2/T) * a0
print('a0', a0)

# Define an, bn
def integrand_an(x, n):
    return f(x) * np.cos(n * 2 * np.pi * x / T)

def integrand_bn(x, n):
    return f(x) * np.sin(n * 2 * np.pi * x / T)

anm = [(2/T) * quad(lambda x: integrand_an(x, i), -T/2, T/2)[0] for i in range(0, n1)]
bnm = [(2/T) * quad(lambda x: integrand_bn(x, i), -T/2, T/2)[0] for i in range(0, n1)]

print('an', anm)
print('bn', bnm)

# Fourier Series function
def fs(x):
    sum = a0 / 2
    for i in range(1, n1):
        sum += bnm[i] * np.sin(i * 2 * np.pi * x / T) + anm[i] * np.cos(i * 2 * np.pi * x / T)
    return sum

y = np.array([f(i) for i in x_])
fss = np.array([fs(i) for i in x_])

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

Fourier series approximation of a sawtooth wave signal without scipy.signal

Ch. 15.3 Sawtooth Wave

Sawtooth Wave

Mathematical Problem Formulation

Sawtooth Wave

Mathematical Problem Formulation